--- kind: concept slug: mathematical-morphology title: Mathematical morphology summary: >- Mathematical morphology uses grey-scale dilation, closing and masked fills, with a fixed kernel per step, to shape an image or depth map. One card: a classical chain that densifies sparse LIDAR depth with no training and about 90 Hz on a CPU. Its constants were tuned greedily on KITTI validation, so they may not transfer. All numbers are author-reported. tags: [image-processing, depth, cpu-inference, no-training] papers: ["1802.00036"] related: [depth-completion, image-denoising, canny-edge-detection] status: draft updated_at: 2026-11-01 --- ## Overview Morphological operations change an image by looking at a small neighbourhood around each pixel, called the kernel, and combining the values in it. The paper's pipeline uses OpenCV's grey-scale operations, in which dilation keeps the larger value in the neighbourhood, so a bright value spreads into its neighbours [arxiv:1802.00036 §4]. Closing joins small gaps by combining dilation with erosion, and a masked fill writes values only into pixels that were empty [arxiv:1802.00036 §4]. The shared idea is that the operation is fixed by a kernel shape and size, and that the order of the steps matters. The paper's central point is that the order needs the values to be inverted first: otherwise the empty value zero would overwrite real depths in a dilation [arxiv:1802.00036 §4]. The variants below differ only in the kernel and in which optional steps run. The one card for this concept is [Fast Depth Completion on the CPU](/papers/1802.00036), which uses the operations to fill sparse LIDAR depth [arxiv:1802.00036 §3]. ## Variants ### Inverted-depth dilation pipeline (paper's method) Eight steps in fixed order: invert depth, dilate with a 5×5 diamond kernel, close with a 5×5 full kernel, fill empty pixels from a 7×7 full dilation, optionally extend the top of the frame and fill large holes from a 31×31 full dilation, apply a median and a Gaussian blur, then invert back [arxiv:1802.00036 §4]. On the KITTI depth completion test set it reports 1288.46 mm RMSE and 302.60 mm MAE, with a runtime of 0.011 s on an Intel Core i7-7700K [arxiv:1802.00036 tab. 1]. Card: [Fast Depth Completion on the CPU](/papers/1802.00036). ### Optional extension and large-hole fill The top-of-frame extension copies the highest valid value in each column upward, and the 31×31 fill then closes the rest. The authors present both as steps of the algorithm [arxiv:1802.00036 §4], but they also say the two steps add many extrapolated points, and suggest turning them off for practical use [arxiv:1802.00036 §5.2]. On the validation split, the paper's final median plus Gaussian blur gives 1350.93 mm RMSE [arxiv:1802.00036 tab. 3]; the table does not split out the effect of the extension alone. Card: [Fast Depth Completion on the CPU](/papers/1802.00036). ## When to use which | Situation | Use | Why | |---|---|---| | Sparse depth, CPU only, no training data | Inverted-depth dilation pipeline | Runs at about 90 Hz on an i7-7700K and needs no GPU or training set [arxiv:1802.00036 §5.1] [arxiv:1802.00036 tab. 1] | | Sensor or depth range far from KITTI | Re-tune every kernel and the 100 m constant | Constants were chosen greedily on KITTI validation: size, then shape, then blur [arxiv:1802.00036 §5.2] | | Downstream use needs no invented geometry | Turn off the extension and the 31×31 fill | The authors say the extension adds extrapolated points and suggest disabling it [arxiv:1802.00036 §5.2] | | Need the lowest iRMSE on KITTI | Not this method | The paper's nearest-neighbour plus CNN baseline has a lower iRMSE (3.25 vs 3.78 1/km), though a higher RMSE [arxiv:1802.00036 tab. 1] | | Measuring the cost of each kernel choice | Validation split | The ablations use the validation split, not the test server [arxiv:1802.00036 §5] | ## Implementation sketch Pseudocode of the paper's pipeline, in the style of the paper cards. Values are in metres, and zero marks an empty pixel [arxiv:1802.00036 §4]. ```pseudocode # Inverted-depth morphology for sparse depth [arxiv:1802.00036 §4] # Input: depth (H, W) in metres; 0 marks an empty pixel # Output: depth (H, W) in metres def complete(depth, max_depth=100.0, extend=False): # Invert so that the grey-scale max keeps real values [arxiv:1802.00036 §4] valid = depth > 0 depth = where(valid, max_depth - depth, depth) # Dilate with a 5x5 diamond kernel [arxiv:1802.00036 §4] [arxiv:1802.00036 tab. 2] depth = dilate(depth, diamond_5x5) # Close small gaps with a 5x5 full kernel [arxiv:1802.00036 §4] depth = close(depth, full_5x5) # Fill only the empty pixels from a 7x7 full dilation [arxiv:1802.00036 §4] empty = depth == 0 depth[empty] = dilate(depth, full_7x7)[empty] if extend: # Copy the highest valid value of each column upward [arxiv:1802.00036 §4] depth = extend_to_top(depth) # Fill the rest of the empty pixels from a 31x31 full dilation [arxiv:1802.00036 §4] empty = depth == 0 depth[empty] = dilate(depth, full_31x31)[empty] # Remove outliers with a 5x5 median, then smooth with a 5x5 Gaussian [arxiv:1802.00036 tab. 3] depth = gaussian(median(depth, 5), 5) # Undo the inversion [arxiv:1802.00036 §4] valid = depth > 0 return where(valid, max_depth - depth, depth) ``` Points to check when porting: the paper's own output formula subtracts from 100 m for every pixel, while the released code keeps empty pixels at zero [arxiv:1802.00036 §4]. The card records this and other code-only details [Fast Depth Completion on the CPU](/papers/1802.00036). ## Contradictions Within the one paper, the authors describe the extension and large-hole fill as steps of the algorithm [arxiv:1802.00036 §4], and later suggest turning them off for practical use because they add extrapolated points [arxiv:1802.00036 §5.2]. Both statements are the paper's; the page does not resolve them. No contradictions between papers are known, because only one card exists. ## Papers - [Fast Depth Completion on the CPU](/papers/1802.00036): classical morphology and blur chain for sparse LIDAR depth, with no training, author-reported 1288.46 mm test RMSE on KITTI.